Curriculum  /  Math  /  10th Grade  /  Unit 1: Constructions, Proof, and Rigid Motion  /  Lesson 3

Constructions, Proof, and Rigid Motion

Lesson 3

Math

Unit 1

10th Grade

Lesson 3 of 19

Objective

Construct, bisect, and copy an angle.

Common Core Standards

Core Standards

  • G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
  • G.CO.D.12 — Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Foundational Standards

  • 7.G.B.5

Criteria for Success

  1. Define angle as the amount of turn between two rays that share a common endpoint. This endpoint is known as the vertex of the angle.
  2. Use the notion that an angle is made up of two rays with a common endpoint and a specified rotation to construct a congruent angle.
  3. Bisect an angle using constructions and describe that by performing this construction, you divide an angle into two congruent angles.

Tips for Teachers

  • The following resources are helpful to get a visual representation and a written description of bisecting an angle using constructions.
  • Math Open Reference "Copying an Angle" is helpful to get a visual representation and a written description of copying an angle using constructions.
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Anchor Problems

25-30 minutes

Problem 1

Bisect the following angle using constructions. Write your steps for construction.

$${\overline{CA}\cong \overline{AB}}$$

Guiding Questions

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Problem 2

You and your partner will be provided with a list of steps (in random order) needed to copy an angle using a compass and straightedge. Your task is to place the steps in the correct order, then follow the steps to copy the angle below.

Draw circle $$B$$: center $$B$$, any radius.
Label the intersections of circle $$B$$ with the sides of the angle as $$A$$ and $$C$$.
Label the vertex of the original angle as $$B$$.
Draw $${\overrightarrow{ED}}$$.
Draw $${\overrightarrow{EG}}$$ as one side of the angle to be drawn.
Draw circle $$F$$: center $$F$$, radius $$FA$$.
Draw circle $$E$$: center $$E$$, radius $$EA$$.
Label intersection of circle $$E$$ with $${\overrightarrow{EG}}$$ as $$F$$. 
Label either intersection of circle $$E$$ and circle $$F$$ as $$D$$.

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 1 > Topic A > Lesson 3 — Mathematical Modeling Exercise 2

Geometry > Module 1 > Topic A > Lesson 3 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Target Task

5-10 minutes

(Teacher to provide diagrams.) 

  1. Prove that angle____ is bisected by ray ____ using constructions. 
  2. Below is a list of constructions to create a bisected line segment. Put these constructions in order and provide the reason for each of these, and then describe if the list of constructions helps you prove that the line is bisected. 
  3. Copy the angle shown below.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems: 
    • Incorporate algebraic reasoning. Give students two different expressions for each half of the angle bisection. Ask students to determine the sum of the two parts of the bisected angle.
    • Extension to have students think about constructing a square using angle bisectors of straight angles.

Next

Construct a perpendicular bisector.

Lesson 4
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Constructions of Basic Geometric Figures

Topic B: Justification and Proof of Angle Measure

Topic C: Translations of Points, Line Segments, and Angles, and Parallel Line Relationships

Topic D: Reflections and Rotations of Points, Line Segments, and Angles

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